Multi-dimensional Burgers equation with unbounded initial data: well-posedness and dispersive estimates

The Cauchy problem for a scalar conservation laws admits a unique entropy solution when the data u 0 is a bounded measurable function (Kruzhkov). The semi-group (S t) t≥0 is contracting in the L 1-distance. For the multi-dimensional Burgers equation, we show that (S t) t≥0 extends uniquely as a continuous semi-group over L p (R n) whenever 1 ≤ p 0, S t actually maps L p (R n) into L q (R n). These results are based upon new dispersive estimates. The ingredients are on the one hand Compensated Integrability, and on the other hand a De Giorgi-type iteration